In about a year, I think I will be starting my undergraduate studies at a Dutch university. I have decided to study mathematics. I'm not really sure why, but I'm fascinated with this subject. I think William Dunham's book 'Journey through Genius ' has launched this endless fascination.

I can't wait another whole year, however, following the regular school-curriculum and not learning anything like the things Dunham describes in his book. Our mathematics-book at school is a very 'calculus-orientated' one, I think. I don't think it's 'boring', but it's not a lot of fun either, compared to the evalutation of $\zeta(2)$, for example. Which is why I took up a 'job' as as a tutor for younger children to help them pass examinations. I wanted to make money (I've gathered about 300 euros so far) to buy some new math-books. I have already decided to buy the book ' Introductory Mathematics: Algebra and Analysis' which should provide me with some knowledge on the basics of Linear Algebra, Algebra, Set Theory and Sequences and Series. But what should I read next? What books should I buy with this amount of money in order to acquire a firm mathematical basis? And in what order? (The money isn't that much of a problem, though, I think my father will provide me with some extra money if I can convince him it's a really good book). Should I buy separate books on Linear Algebra, Algebra and a calculus book, like most university web-pages suggest their future students to buy?

Notice that it's important for me that the books are self-contained, i.e. they should be good self-study books. I don't mind problems in the books, either, as long as the books contain (at least a reasonable portion) of the answers (or a website where I can look some answers up).

I'm not asking for the quickest way to be able to acquire mathematical knowledge at (graduate)-university level, but the best way, as Terence Tao once commented (on his blog): "Mathematics is not a sprint, but a marathon".

Last but not least I'd like to add that I'm especially interested in infinite series. A lot of people have recommended me Hardy's book 'Divergent Series' (because of the questions I ask) but I don't think I posess the necessary prerequisite knowledge to be able to understand its content. I'd like to understand it, however!

Please take what Harry says with a grain of salt,Max.He thinks anything with motivation is not mathematics and that's not good for beginners.
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The MathemagicianJun 14 '10 at 21:10

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Max, I think that the sooner you stop worrying about questions like "But isn't Topology more of a graduate subject?", the better. Most Bourbaki's books do not make good first reading for the subject, that's true, but there are topology books that can, and should be, read while still undergraduate. There is no such thing as undergraduate/graduate subject, there is mathematics and something else.
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Vladimir DotsenkoJun 14 '10 at 21:34

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1. Some of the answers you have received are a little odd, IMO. I see some fairly difficult graduate level books being recommended for a high school student who knows some calculus. (Homological algebra? Really?) I honestly don't know what to make of this. 2. Generic advice that may or may not be something you need to hear: Don't get discouraged. Math is hard for everybody. Be persistent, but if you have a book you can't make progress on, don't feel the least bit of shame in turning to a more elementary treatment or going back to learn prerequisite topics or whatever it takes.
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Mike BenfieldJun 14 '10 at 22:10

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Aack! If you want italics, on MO put underscores _ or asterisks * on either side, not dollar signs. In TeX, use {\em text }. Dollar signs make the computer process whatever's inside as math, as if you had all those variables to multiply together. The classic example is $difference$ versus difference — notice the spacing around the f s. (In the default TeX font, the correct look is $\textit{difference}$.) The spacing is even weirder for words with ffi: $spiffier$, $\textit{spiffier}$, spiffier.
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Theo Johnson-FreydJun 14 '10 at 22:10

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I looked at the table of contents for the book the OP linked to ("Introductory Mathematics:...") and they define things like sets, functions, injective, bijective, complex numbers, vector spaces, etc. I think it would be considerate if people kept this in mind when suggesting books. Some people have not done this, and I would agree with Mike Benfield that you shouldn't be discouraged if a randomly chosen book from the answers is too difficult to understand right now. Many of them will still be difficult after several years of studying math.
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Peter SamuelsonJun 15 '10 at 0:33

31 Answers
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I'm not a big fan of full roadmaps and reading lists. Exploring mathematics is something that can be totally different depending on where and who you are. Any serious roadmap needs to be flexible and take account of the course: reading maths is a skill (one you seem well on your way to learning btw! but still...), initially you may find actual teaching easier to grasp- and your reading should work along side that. So here's my attempt at a flexible roadmap:

1) Buy some very carefully chosen books and read them cover to cover: There's a lot of baffling books out there- even some that look really UG friendly can have you weeping by page 5 in your first year- and you only want 4-5 to start with (any more will just be too expensive and you won't get round to reading all of them- top up via the library). My recommendations are: 'Naive set theory- Halmos', 'Finite dimensional vector spaces- Halmos', 'Principles of mathematical analysis- Rudin' and 'Proofs from the book- Aingler and Zeigler'.

[These, ostensibly, cover the exact same material as the book you have decided to buy- but to develop quickly I urge you to buy more mathematical texts like these: Halmos' and Rudin's writing styles are very clear but technical (in a way that the book you are interested in will not be), and will make you a better mathematician faster than any book that tries to 'bridge the gap' ever would. I also seconded Owen's call for proofs from the book: it is simultaneously inspiring and useful as a way of seeing 'advanced' topics in action- it's something you'll keep coming back to, right up to your third year!]

2) Do all of the excercises: Or as much as you can bear to- even if it looks like it's beneath you (if you're half decent- a lot of first year will!) you will be surprised as to how much it helps with your mathematical development (and the crucial high mark you'll need for a good PhD placement). This applies to classes and your 4-5 text books.

3) Ask your tutor about doing some modules from the year above: If you've read all of those textbooks and done all of the excercises, you will be ready. Get some advice from your tutor about what would be best and roll with it (most unis won't make you take the exam, so if you don't feel comfortable you're fine). Taking something like metric spaces or group theory in your first year will put you top of the pile.

4) Keep doing all of these things: Immerse yourself in maths- keep on MO, meet likeminded people and no matter how slow the course seems to be moving, no matter the allure of apathy: keep at it. Advice for later books would be pointless now, but there will be people who can give it to you there and then (use maths forums if you want). Oh, and never rule out an area- you never know where intrigue will come from...

I would recommend Courant and Robbins' "What is Mathematics." It is quite inexpensive, and gives a meaningful introduction to many areas of mathematics. I think you can browse the table of contents on Amazon.com to see what is included.

There are lots of good answers here, so I'm not going to add any additional book recommendations. I just want to warn you of one misconception I had when I was in your position. It is best illustrated by example but don't worry if you don't understand all the terms. That's part of the point.

A vector space is mathematical structure defined in terms of another called a field: for example the real numbers are a field and the plane is a vector space. Now, a field is a special case of another structure called a commutative ring. A field is just a commutative ring in which you can do division; the integers are an example of a commutative ring. Now, commutative rings are built out of abelian groups, which are themselves a certain kind of group.

My reaction to seeing such definitions was to assume that the best way to learn about the ones at the top of the hierarchy (e.g. vector spaces) was to develop a solid understanding of those at the bottom (e.g. groups). This seems natural because math is supposed to be a very methodical thing and logically if B is defined in terms of A you might expect you'd want to understand A first.

It turns out that this is for the most part wrong. The reason is that there are all sorts of crazy groups out there, but the abelian ones are some of the simplest and easiest to understand. This type of reasoning can be applied at each level, and when you get all the way up to vector spaces, you get a family of objects which behave very nicely, having eliminated some complicated behavior at each stage.

Of course, you won't be able to appreciate quite how nice the situation is until you later on learn what can go wrong when you take a few steps down the hierarchy. But generally speaking, it is easier to learn about objects with lots of structure than those which have very little.

Definitely agree. But, at the same time, there is a sense in which additional generality makes things easier to understand (e.g. defining Lebesgue integration via abstract measures instead of using step functions and monotone convergence as it was done before abstract measure theory was developed). So I think it goes both ways.
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Akhil MathewJun 15 '10 at 23:39

Forget buying books. Just buy access to a good university library. That will be much cheaper. Make sure it also gives you access to online journals.

I would suggest starting with group theory, analysis(real and complex), elementary number theory, algebraic number theory, analytic number theory and then working your way up to algebraic geometry, differential geometry and so on.

I found the books published by the Mathematical Association of America very useful to get some intuition.

You may also want to browse through the American Math Monthly. It has lots of tutorial-style articles on various subjects. Search mathscinet for articles which have won expository writing awards. For example, I found this article quite useful

I agree with Andrew Stacey. That's a cool diagram, but it's from the viewpoint of physics as the only end goal.
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Chris PhanJun 15 '10 at 10:56

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Buying access to a university library is a good idea, but may not take the place of buying some texts. In the university libraries of my experience, the classic texts discussed in the other answers would be perpetually checked out.
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Nate EldredgeJun 15 '10 at 23:46

I am a current undergraduate student and remember the feeling of wanting to know more before starting my degree. My university sent out a letter recommending that we all buy a certain analysis text before the start of the course, and that if we wanted to do some work before term that this was the book to read... Suffice it to say that the book was awful, and not even particularly useful once I had started at the university... a complete waste of my time and money.

Whilst this is slightly biased towards my own interests, if you are interested in combinatorics at all then I fully recommend 'Combinatorics and Graph Theory' by Harris, Hirst and Mossinghoff. The book is certainly readable for an enthusiastic school student, and really covers the basics of combinatorial enumeration and selected topics in graph theory. What's more the book is written with great clarity, and a good sense of humour. It starts at a basic level, but works up to material that there was not room to fit into a term long second year module.

Another text that I cant speak more highly of is 'Proofs from the Book' by Aigner and Ziegler. This is a compendium of interesting and simple proofs taken from many areas of mathematics. The title refers to a line of Paul Erdos. To Erdos, a book proof was a beautiful, elegant solution to a problem... so that is exactly what you find in here. Not all of it will be accessible immediately, but a lot will, and by the time you finish your first year you should be equiped to understand all of it.

Finally, from a softer approach: Martin Gardner's columns from Scientific American have been collected into several volumes which are readily available. Gardner often wrote about pure mathematics but also sometimes about philosophy of science, or physics. The articles are written so that the reader needs no technical knowledge, but I still read through the books just to find out a little bit about something that I haven't studied. Even if it is something that I do know about, reading Gardner is still a great exercise in seeing how good teachers really work: and there can be no doubt that Gardner was an excellent educator.

I second the recommendation of Proofs from the Book.
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Chris PhanJun 15 '10 at 10:59

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@Owen I'm VERY glad someone finally recognized the wonderful text by Harris,Hirst and Mossingoff on a very baffling subject. It is simply one of the most readable and perfectly structured introductions to any subject I've EVER seen. After reading this book and absorbing it's contents,the student will be more then prepared to move on to more advanced texts on the subject like Gross and Yellen's GRAPH THEORY WITH APPLICATIONS (the bible of the subject,in my opinion) and Bona's texts on combinatorics.
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The MathemagicianJun 15 '10 at 13:46

Here are a few introductory books that I've found useful. I'm in a somewhat similar situation as you, I suppose--a year older, but having had to learn whatever mathematics that I know mostly independently (though I've been lucky to have been able to talk to several mathematicians on various occasions as well as take a couple of courses). For more advanced ones, I'd certainly be the wrong person to ask. Anyway, here are some of the books that I've used (or am currently using).

Algebra: I think Herstein's Topics in Algebra is often recommended as a good introductory-level undergraduate textbook (which I learned material from). Lang's Algebra is a nice follow-up, but I think a lot of people (certainly including myself) would be turned away from the subject if Lang were the first thing they opened.

Analysis: Rudin's Principles of Mathematical Analysis is great, but the real fun is in Real and Complex Analysis (you don't have to read all of the former to get into the latter). Also, Peter Lax's Functional Analysis is very enjoyable reading.

Differential geometry: I personally found it very hard to read Spivak's Calculus on Manifolds because there wasn't much motivation (like, why would one define a tangent space when you're just working with open subsets of $\mathbb{R}^n$?), but volume 1 of his A Comprehensive Introduction to Differential Geometry has similar material with much more explanation and motivation.

Number theory: My first introduction to number theory was from Niven and Zuckerman's book. Serre's A Course in Arithmetic is also very interesting, but I would consider that more of a second or third read (I tend to find Serre's books rather terse and consequently struggle with them, but this is likely just a personal failing). For instance, you need to know what a projective limit is to read it.

I wish I knew a really good introduction to algebraic number theory. The ones I've seen have tended to be somewhat difficult (i.e., presupposing a fair bit of material in Lang's algebra such as familiarity with localization, noetherian rings, etc.--topics that might be omitted in a first course on abstract algebra). I think Lang's Algebraic Number Theory is a great book, but it took me an enormous amount of time before any of it started to make any sense at all. And there are other "elementary" books on algebraic number theory that never get anywhere interesting.

Ireland-Rosen's A Classical Introduction to Modern Number Theory has lots of fun stuff in it and is a great book to read after one understands elementary abstract algebra (hat tip to Vladimir Dotsenko for pointing this out; I had read the book and then forgotten about it). There is also a little bit of algebraic number theory in it.

Linear algebra: I learned this from Hoffman and Kunze's book, but I think the subject is nicer in a more abstract context (e.g. after one has talked about rings).

Computer science: Wait, hold on! I think I'd be remiss if I didn't mention Sipser's An Introduction to the Theory of Computation, which has to be one of the most enjoyable books I've ever seen---and it is basically mathematics.

Logic: Ebbinghaus, Frum, and Thomas have a very nice book on mathematical logic in the UTM series.

Topology: Dugundji's book is a bit old, but I found the exposition very crisp and enjoyable. At the same time, I suppose it's not good for geometric intuition. I have heard great things about the books by Munkres but have not (yet) read them myself.

The books in the Carus mathematical monograph series are all accessible, pithy, and enjoyable; Krantz's Complex Analysis: The Geometric Viewpoint is one that I'm enjoying looking at right now.

Another bit of advice: You don't have to finish a book to "graduate" to another one! Skipping around is something many mathematicians do, and one never "really" understands an area of mathematics (at least, not as an undergraduate), so it's more efficient to move on to other things. Nor do you have to know everything. (This isn't me giving advice; it's me regurgitating something that a very respected mathematician told me a while back, and was quite a revelation for me.)

I obviously don't know whether you're near a university library and have borrowing privileges; I happen to live near three (albeit ones from liberal arts schools with no graduate math department). If not, I strongly recommend buying used books, since math textbooks tend to be ridiculously overpriced for some reason. Fortunately, there are many good resources on the web: James Milne's site is excellent, for instance.

Anyway, if you need more, the bibliography on my blog has a longer list (but the ones here should probably keep you busy for at least a little while). You could also try contacting a professor at a nearby university to see if he or she is willing to mentor you for a research project; there may be programs through which this is possible (though admittedly I have no idea how it works in Europe). The benefit of this is that you'll end up absorbing a lot of new mathematics along the way as well as better understanding what you already know.

There's an old book by Hasse on algebraic number theory, I think, which is very well written. Also, "A Classical Introduction to Modern Number Theory" by Ireland and Rosen is a good read.
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Vladimir DotsenkoJun 14 '10 at 22:50

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Of course! How did I forget Ireland-Rosen?
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Akhil MathewJun 14 '10 at 23:00

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Samuel's book "Introduction to the algebraic theory of numbers" is an excellent first book on algebraic number theory: rather self-contained, good algebra technique, excellent examples/applications worked out, proves some real finiteness theorems, and nice (but slightly small) selection of exercises at the end.
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BoyarskyJun 15 '10 at 0:19

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I second Boyarsky - we used it for a semester undergraduate course in algebraic number theory (we finished the book about 2/3 through the semester, then talked about L-functions and proved the Chevotarev Density Theorem). Another good book with slightly more problems and material than Samuel is Marcus's Number Fields. And if you really feel that both of those "never get anywhere interesting," I recommend Number Fields by Janusz as a second read. You can skip most of the first four chapters if you've done Marcus, and Janusz discusses class field theory.
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David CorwinJun 16 '10 at 14:54

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Thanks again Akhil (my comment was removed for some mysterious reason). Perhaps we meet each other one day, as 'true' mathematicians ;).
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Max MullerSep 24 '10 at 23:19

Algebra/Linear Algebra: A Course In Algebra by E.B. Vinberg (gives an integrated introduction to both linear and abstract algebra with lots of applications,especially to geometry:VERY readable and reaches a very high level of coverage by the end), Michael Artin's Algebra (more intense,but similar in spirit,a second edition coming out later this year,can't wait for it)

Topology: A First Course In Topology:Continuity And Dimension by John McCleary (the best introduction to the subject)

Overview of Mathematics: Mathematics:It's Content,Methods And Meaning by A.N.Kolomogrov,etc. (The great classic by 3 Russian masters of mathematics,will give you the best bird's eye view of the subject)

That should get you started and give you the tools needed to go forward. Good Luck and welcome to the sorcerer's guild!

Max, if you like pictures, check out "A topological picturebook" by Francis, it's quite good and informative!
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Vladimir DotsenkoJun 14 '10 at 21:54

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@Max: EVERY mathematics book for beginners should have pictures and don't let anyone try and make you feel stupid for asking that. Human learning is largely spacial in nature. A picture is not a proof-but a proof or concept can be made much more understandable with a picture accompanying it. This is ESPECIALLY true in topology and geometry.
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The MathemagicianJun 14 '10 at 22:01

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Re: pictures. I would say that not just "EVERY mathematics book for beginners" but indeed every mathematics book should be replete with pictures. Humans think well visually. But note that pictures have at various times been antithetical to mathematical writing. Euclid and the other Greeks certainly drew pictures when figuring out their work, but their writing are completely devoid of them, because a picture is not a proof, which should apply to any (theoretically perfect) geometric construction, not to the (necessarily imperfect) representation in a given picture.
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Theo Johnson-FreydJun 14 '10 at 22:14

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Re whether every math book should be replete with pictures: Pictures often help, but I would not go so far as to say that they always help. Sometimes the movie adaptation of a novel is better than the novel. But sometimes the novel is better -- it might, for example, "leave more to the imagination". Re whether a picture is a proof: Try asking Rob Kirby whether a picture is a proof.
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Kevin H. LinJul 20 '10 at 9:10

Abstract Algebra by Dummit and Foote: I have heard many people speak ill of this text. However, from the perspective of one who is interested in picking up the fundementals of algebra, in my opinion this is the text.

Introduction to Topological Manifolds by Lee: Don't let the word manifold scare you off. This book is essentially an introductory topology book with all of the boring bits left out. I know it is a GTM but I was able to pick it up during my first year of Uni and learn a lot of topology from it.

Munkres topology: This is "the" book for learning point set topology. It is a bit gentler than Lee's and goes through all the relevant set theory you will ever need (unless you become a set theorist). I picked this book up during my first year at Uni and have been opening it regularly ever since.

I also recommend Principles of Mathematical analysis by Rudin. It has already been mentioned though. It is an amazing book. It is also the only book which i have ever really studied from start to finish.
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Daniel BarterJun 14 '10 at 22:16

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I don't think Max here is an ordinary student. He seems like he is interested in becoming a mathematician, and he seems very bright. In this light, even if he is not able to handle Lee's topological manifolds straight away, I would be very surprised if he cant handle Rudin's Principles of Mathematical analysis. Once he has mastered this book, he should be able to move onto Lee's book easily.
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Daniel BarterJun 14 '10 at 22:29

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@ Andrew L: Rudin was my first introduction to analysis. In hindsight, i knew what a derivative was (but not to much more) and i could mechanically evaluate integrals from high school. Max seems to know a lot more than I did when I first picked up rudin, and I didn't have to much trouble working my way through it. I am not telling him that he should read rudin, I am just telling him that I think he would be able to handle it if he wanted to and I think it is a good book.
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Daniel BarterJun 14 '10 at 22:35

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@Andrew L: I get the feeling that you've never even read Rudin or Bourbaki.
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Harry GindiJun 15 '10 at 5:02

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Andrew, chill out. Different people have different tastes.
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Kevin H. LinJul 18 '10 at 8:01

Great introduction to various ideas of algebra and topology: Abel's theorem in Problems and Solutions by Alekseev. Another advantage of this book, as well as Halmos "Linear Algebra Problem Book" mentioned elsewhere in answers is that it gives a balanced set of problems - and one of the most efficient ways of learning new things is through problem solving.

@Vladimir Lang's Algebra is WAY too difficult for a beginner even at Moscow State. I'm sure most students that tried to use it struggled. Frankly,most of the Russian trained mathematicians I know learned algebra from Kostrikin's books.
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The MathemagicianJun 14 '10 at 21:58

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Andrew, I believe that those Russian trained algebraists whom I know who did not learn from Lang (and they are minority), learned algebra from Vinberg's book. In any case, there is a possibility that we know different mathematicians, and don't intend to question your statement. I hope that you don't intend to question mine either.
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Vladimir DotsenkoJun 14 '10 at 22:05

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I think that for someone who has not studied abstract algebra, the book by Dummit and Foote is much better than Lang. They make a much more concerted effort to explain and motivate concepts, which I think is important when first learning a subject.
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Peter SamuelsonJun 15 '10 at 0:25

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@ Vladimir I'm not-and I probably know mathematicians from a previous generation. Vinberg's text is my algebra reference of choice and I am SO glad it's available now to an English-speaking audience!
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The MathemagicianJun 15 '10 at 4:07

There are (today) many popular books on mathematics. I don't know any very good ones. Kac and Ulam, Mathematics and Logic, is an older style of popular book, pretty good I think, and on Google Books. (I'd be embarrassed if I didn't at least mention Timothy Gowers, Mathematics: A Very Short Introduction.)

There are plenty of textbooks, and books put together from lecture notes. These are mostly not so exciting.

You really need a book on analysis, one on algebra, and one on geometry or topology. These are for core ideas. Fourier Analysis by Thomas Körner is mostly on Google Books: Körner is the right kind of writer for your requirements. I learnt a great deal from Serge Lang, Algebra, an early edition. But his writing now seems strange to me - bias towards number theory, which is OK really, and later editions are very different. Geometry and topology are hard. Maybe Penrose's book on everything might help with concepts (bias towards physics). Any book that helps you grasp what a Lie group is. None of these writers are "orthodox".

Classics: Hardy's Divergent Series you mention, but it is too hard to read, maybe few really can these days. Weyl's Classical Groups has always been impossible? Serre, A Course in Arithmetic, yes.

@Charles Halmos is a wonderful author and you made me realize the one book I should recommend to everyone thinking a career in math:I WANT TO BE A MATHEMATICIAN:AN AUTOMATHOGRAPHY. A MUST READ.To be honest,though-I think between the 2 texts on linear algebra he wrote;FINITE DIMENSIONAL VECTOR SPACES and A LINEAR ALGEBRA PROBLEM BOOK,the latter would be much easier and enjoyable for a beginner. FDVS is very hard and abstract.
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The MathemagicianJun 14 '10 at 22:06

The following texts were very successful in helping me make the transition from high school mathematics to university level mathematics. When I left high school in the UK system (A-levels) I had a reasonably thorough grasp of calculus, trigonometry, geometry, school level algebra, statistics and mechanics (these were covered in the A-level maths and further maths syllabi, if you're familiar with these at all).

General reasoning:

Copi and Cohen, 'Introduction to Logic'. A basic introduction to logical reasoning. Despite having studied as much mathematics as I could at high school, I was never taught to understand the logical structure of proofs. Reading this book helped me to begin reading proofs in undergraduate/graduate mathematics books.

Analysis:

Bartle and Sherbert, 'Introduction to Real Analysis'. This is useful for picking up the basics of real analysis at an elementary level and is also useful in learning to read proofs of elementary results. I found that this phase of my education was a bit tedious and that I was wanting to get ahead to a more advanced text. It is a useful complement to item 3 below.

Kolmogorov and Fomin, 'Introductory Real Analysis'. This is an excellent book for learning analysis and was used in some honors level analysis courses for students who were familiar with reading and writing proofs. I would emphasize that reading Copi and Cohen first is an imperative.

Walter Rudin, 'Principles of Mathematical Analysis'. Harder to read than Kolmogorov and Fomin, but it contains beautifully written proofs and is a great text to model your own answers on.

T. Gamelin, 'Complex Analysis'. A good book on complex analysis with lots of motivating examples.

Algebra:

Allan Clark, 'Elements of Abstract Algebra'. This is a Dover publication and is rather cheap (probably around $15 or so now). It consists of a hundred or more articles; you are given definitions and the proofs of a few important theorems. Everything else is an exercise. I believe this book did the most for me in helping to build intuition for abstract algebra.

Hoffman and Kunze, 'Linear Algebra'. A classic book on linear algebra which I do not think has been surpassed. It provides thorough and well written proofs. I believe this is a preferable text to Axler's book, unless you are somewhat heavily inclined towards analysis and/or do not enjoy doing computations.

Serge Lang, 'Algebra'. This was my second book in abstract algebra, after Clark's. It is beatifully written, which is why I prefer it to Hungerford's book. As another answerer mentioned, the typesetting in Hungerford's book is also somewhat off-putting, and Lang's book does not suffer from this defect.

Topology:

Munkres, 'Topology'. A clearly written text with a good supply of examples.

Lastly, I would second the idea of purchasing access to a university library with a good collection of mathematics textbooks.

It is a translation from the german book "Zahlen", which I read with pleasure during my first year at the university. All authors are german mathematicians known for their wonderful writing.

The book teaches you a lot about different number systems, like the complex numbers, the Cayley numbers, nonstandard numbers and so on. It also connects these topics to other topics in mathematics, so it provides a beautiful example-based introduction to mathematics. It doesn't stop at basic material, so you can revisit this book after one or two years of university courses, and with a new perspective, learn something new again!

I have to warn you that it's not a book about number theory, although it touches elementary number theory preliminaries at various places and I guess you should read something like it before reading any serious number theory.

The book is full of historical remarks and motivational paragraphs. Nevertheless, it's written in a clear and formally correct way (not the sometimes difficult "colloquial style" found in American math introduction books).

If you're interested in infinite series, this book might be especially good for you.

I think some of these books are at a bit high of a level. I can't imagine anyone opening "A course in arithmetic" without a good grounding in Both Algebra and Complex analysis.
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Daniel BarterJun 14 '10 at 22:09

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I looked at Milnor's "Morse Theory" and there's a good chance that the following words, which appear on page 1 but are not defined, will be foreign to someone who hasn't studied much beyond caluclus: manifold, torus, tangent, homeomorphic, 2-cell, cylinder, compact, genus, and boundary. This book is clearly not appropriate. I would have similar reservations about recommending several of the other books.
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Peter SamuelsonJun 15 '10 at 0:15

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"I think it's worth mentioning that reading all of these books would take most people most of their time as an undergrad." - you must be kidding. I mean, it can be true if you include in "most" all people who have no intention of becoming a mathematician...
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Vladimir DotsenkoJun 15 '10 at 18:29

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Hmm... I wonder how many people "who have intention of becoming a mathematician" have completed Hatcher's Algebraic Topology. Milnor's Morse theory is much more elementary than that, not to mention less monumental.
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Victor ProtsakJul 18 '10 at 14:37

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@Ben Crowell: Disservice to whom? I fail to see how such information could be relevant to the original question (“what should I read next?”). There is no way to classify these links according to you scheme without emailing each author separately, not to mention that Rudin, Kostrikin, and Cartan are no longer alive, hence for their books it's impossible to find out the answer. Finally, it seems that the value of such classification is close to zero anyway (how one could potentially use it?). Also, being in copyright (2 vs 1 and 3) depends on the jurisdiction, of which there are about 200.
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Dmitri PavlovOct 23 '12 at 17:24

Studying mathematics for the sake of learning is a feat worthy of praise; studying mathematics to fulfill a particular goal (such as gaining an academic job, solving problems in industry and society, or using it to further research in other disciplines) is a feat that is worthy not only of praise, but of asking others for financial support to help one do it. You can budget time for doing the first as long as you leave time for
earning the money to pay the bills. You can budget a little more time for doing the
second as long as you leave time for getting the monetary support, and it is important that you pick a good goal: a goal that others are willing to support.

Even if you do not have to support yourself, it is good to have a goal and a plan around
mathematical study. One plan might be to spend x amount of time reading reviews and
abstracts, and then use that to pick papers and books to spend y amount of time beginning to read. If you develop questions and sufficient maturity, some well placed questions here on Math Overflow may supplement your reading. Part of doing mathematics well is asking the right questions.

I recommend using as many free resources as you can before spending money on books. The
Internet, your prospective university library, and friends and acquaintances who have
books of interest are just three sources. Don't worry about the sophistication or the
prerequisites needed for understanding a book or article: you are compiling a list of
things for future reading and enjoyment. Buy something only for adding to your "permanent library", the library you will always carry with you or will always devote time and
energy to store and maintain, or buy something only when you know you will give to someone
else but you will get to read it for yourself first.

Finally, supplement your study with attending lectures on other topics. The engineering or computer science departments (or other academic departments even) will sponsor lectures which may supplant your current interest in mathematics. Things get exciting when you
discover an application of what you learned to another field, but you won't see the applications without seeing something of the other fields.

This is a tricky question to answer because I'm not sure what your current mathematical background is and what exactly you want to achieve. Are you familiar on how to write proofs? Given a statement, can you take its contrapositive? Or its negation? Do you know modular arithmatic? (I'm a bit confused with what you're asking for because the first book you want to purchase is an introductory text. Then you ask for graduate level reading material.) Anyways, if not, then you might want to try a little book by Peter Eccles called "Introduction to Mathematical Reasoning." It's a book that is designed to help transition students into their first higher mathematics course. He goes over all the fundamentals: basic logic, truth tables, set theory, sequences and series, divisbility, modular arithmatic, elementary combinatorics. It's self-contained and very readable. He also provies solutions to many of the exercises as well. I'd definitely start with that text if you want to learn how to write proofs.

After that, then a nice text is the two volume series "Mathematical Analysis" by Vladimir Zorich. It's a very comprehensive and well-written text on real analysis. A nice aspect is that he gives the historical motivations for a lot of constructions so it's easy to understand how the mathematical ideas arose.

If the "two volume" series of text book scares you off, then Stephen Abbott has a nice book called "Understanding Analysis" that I would suggest checking out.

Of course, there are a lot of good suggestions out there. It'd help to have a more concrete idea of what you're looking for.

Fundamentals of University Mathematics by C. M. McGregor, J. J. C. Nimmo, and W. W. Stothers is an excellent book for the transition from high-school to university mathematics. Definitely pitched at an appropriate level and worth a look.

Some may protest that it's not really a math book, but I have to recommend Gödel, Escher, Bach, by Douglas Hofstadter. It is, among other things, a beautifully written and extremely self-contained introduction to three fascinating subjects:

Formal systems. What does it mean to prove a theorem from a set of axioms? How do we translate mathematical statements into statements about the real world? Is it possible to do math without thinking about the real world?

Computability theory. How do you unambiguously describe a computational procedure? What are the basic tools of computation---and what price do you pay for using the most powerful ones? Can the values of a mathematical function always be computed?

Gödel's incompleteness theorems. Can every true statement in the language of number theory be proven from the axioms of number theory? How do we know that the axioms of number theory don't subtly contradict each other? Can a mathematical theory be used to study itself?

Winding its way through these weighty topics, GEB manages to find time for a little mathematical sightseeing, discussing some classic proofs, problems, and pieces of history.

Most importantly, it's also tons of fun to read! ^_^ You'll have plenty of time for dry, boring books at university; in the meantime, you should just read what you enjoy. ~_^

p.s. GEB is a massive book, and I do not recommend trying to read it straight through. If you want to focus on the mathematical parts, you might try concentrating on these chapters:

In high school, I was never really exposed to the kind of books that mathematicians use in their mathematics education, the kind of material that practicing mathematicians assume that "everybody" knows. The resource that I found most helpful for identifying these "canonical" books used in the top American mathematics programs was the Chicago Undergraduate Mathematics Bibliography. It hasn't been updated in some time and was written by people who graduated about 10 years ago, so it is missing some great newer titles like Lee's Introduction to Topological Manifolds and Hatcher's Algebraic Topology. However, it does have a great list of books from the high school level to early graduate level and short reviews of each, all on one page.

As a motivated student interested in mathematics, I can think of no better way to get started now than to read some of the reviews, get the books that sound interesting, and work through them. Most (if not all) of the books listed are classics and thus are familiar to nearly everybody doing research in that particular field. Furthermore, this has the added benefit that these textbooks are often cited for background in papers/monographs/lecture notes and will likely be used as references for classes that you take at university.

Thanks (I think) for the link to the CUMB -- rereading it just now was an interesting experience. Based on a few context clues, most of the text was probably written in late 1997 or early 1998. Perhaps the most dated point is the belief that Massey's GTM is, for better or worse, the standard introductory text in algebraic topology. Nowadays the canonical choice seems to be Hatcher's book, and rightfully so.
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Pete L. ClarkJun 15 '10 at 0:00

I don't know how much sense it makes to study books on the topics of the usual first courses at university since this is stuff one learns anyhow at university. This means not that I recommend to study material on higher courses. I mean that one perhaps should study elementary mathematics, which is not usually covered so much at university, but is worth knowing.

A very fine book on elementary geometry is Coxeter's Geometry Revisited. It covers on 200 pages a lot of beautiful mathematics.

Besides the already mentioned Proofs from the BOOK, I would recommend to study elementary number theory, where one gets with little formal effort quite deep theorems. Books I used (and liked) include A Classical Introduction to Modern Number Theory by Ireland and Rosen and An Introduction to the Theory of Numbers by Niven and Zuckerman, but there are a lot more: just read the amazon reviews. They are usually quite informative. Andreescu has also some quite nice problem based books.

There is also another book I would like to recommend, but hesitate since it is not really easy. It is Thurston's Three-Dimensional Geometry and Topology (Volume I). But on the other hand, it is a book one can learn very much from and it is written in a quite informal way, bringing much ideas, and begins with elementary ideas from two-dimensional geometry. I've given a short series of talks to (bright) high school students on some topics of the first chapter and I think, it was successful, so if you want to try and like to use your geometric imagination, then try.

Niven, Zuckerman is very nice, but there exist so many other, different good books that I recommend to visit a good math library with an experienced mathematician and select in a kind of 'group-browsing' what fits best.
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Thomas RiepeJun 17 '10 at 12:44

Since you live in the Netherlands, you might want to look at what is currently being offered in the Springer Yellow Sale 2010 (http://www.springer.com/sales?SGWID=5-40289-0-0-0). You'll find Lee's Introduction to Topological Manifolds there, for example, or Abbott's Understanding Analysis.

I do not understand why nobody mentioned Schaum's Outline of Theory and Problems. These books offer a lot of practice in different branches of math. From algebra to calculus to differential equations,logic, topology,...

It has always been a great idea to practice my math skills with the problems there.

Another thing, I learned calculus I to III before getting in to univ from the book:

Do you live near a university? Maybe someone in the university's math dept. would be helpfull? Anyway, I'd suggest to get access to it's library, take a few days to find a handfull of texts which look accessable and make you most curious, then work through them. E.g. Kintchin's "Three pearls of number theory" may be an idea, it was written for someone in your situation, but is not easy. (some thoughts on learning etc.: 1 ,2 )

If you are looking for a break from Calculus and don't want to dive into Category Theory right away, my favorite book is Alan Beardon's Algebra and Geometry. What follows is an enhanced Amazon review I wrote as an undergrad. The Cambridge Schedules provide a wonderful guide to further study so do have a look.

Beardon's book has become a bible of sorts to first-year students studying mathematics at Cambridge University (please refer to the Schedules for a beautiful play-by-play of topics and books by perhaps the best foundational math curriculum the world-over). Its quality as a text cannot be doubted, although its usefulness for further years of algebra is limited. This is precisely the book to study from if you are doing vector calculus and differential equations, but still aren't sure about doing mathematics seriously. If you have not taken a course in linear algebra or abstract algebra, buy the paperback copy (~$50) of this book and start reading right away. Beardon starts with (what I believe is the best way) the study of permutations (think about shuffling a deck of cards) to develop an intuition of the basic notions of a group. From here the fundamentals for further study in mathematics is laid. I won't repeat the table of contents here, as you can look for yourself, but believe me when I say that mastering the concepts in this book will serve you very well.

I truly wish I had a course which devoted itself to the complete digestion of this book. I used it for self-study and found that it served me very well. It does not fall easily into the structure of most American math sequences, as these departments are often forced to "modularize" mathematics into semester-bite-sized pieces. I believe that this often has a negative impact on the appreciation of mathematics as a whole, especially at the nexus between doing basic calculus and appreciating proof, rigor and beauty in mathematical structures.

The book may not go into the same depth as, say, Artin's "Algebra", but rather the foundational concepts for the study of algebra and geometry are emphasized in a variety of settings. This is very important as the study of "abstract algebra" is precisely that if you do not have a wide-selection of examples and contexts to draw from. This book has plenty of exercises of varying difficulty, and everything in this book is accessible to the beginning student of mathematics.

Bottom line: If you are someone interested in learning linear algebra, geometry, group theory, Mobius transformations, complex variables all in a rigorous yet introductory level, this is the book for you. Developing a robust mental model for mathematics requires building several thin layers at a time. This means not going too deep too quickly, but rather snorkeling around the entire reef, before you gear up for further exploration.

Beardon's book to me is one of those "best intentions" books that doesn't quite deliver. While it presents a wealth of beautiful material that all math students should know and isn't covered in most traditional sources,the lack of pictures really hurts the presentation. It also doesn't have anywhere near enough examples or exercises. Elmer Rees' classic Notes on Geometry is much better in this regard. Here's hoping that Beardon one day gives us an enlarged second edition that fixes both those defects-if he does,we'll have a bona-fide classic.
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The MathemagicianJul 18 '10 at 20:26

EDIT:some people didn't like what I've written below. Rather than change it, I've added comments below instead. Why should I change it anyway? If you don't agree, that's your opinion; but these are mine, so why shouldn't Max see a fair range of differing views? One of the things I was trying to say is that if you want to have a successful career ACTUALLY USING PROPER MATHEMATICS beyond just, say, calculating percentages or putting numbers into a formula, then your chances really are very slim unless you're EXCEPTIONALLY good. The number of good jobs with genuine mathematical content is far, far smaller than the number of mathematicians with reasonable ability (at least in England). The sooner Max knows this, the better he can prepare properly and take precautions.

Hi Max, it's nice to see some youngsters with interest!

Firstly, Hardy's Divergent Series is a great, fun book, but probably too hard and of little use for your future career (it's very old-fashioned and not likely ever to come back into fashion - but I could be wrong).

My personal opinions for what they're worth (i.e., not very much); some (many?) people would disagree!

The most important thing by far is to go to the very best university you can. I assume that you have genuine mathematical talent (you must be easily the best at mathematics in your school) and that you want to go to the top university in Holland; don't waste your time at second best places. (If not, then improving your school grades or whatever you need is the top priority!) It's far better to be in the middle at the best university than at the top in a poor university.

If you can't decide which is the "best", ask your teachers, see which ones have Fields Medallists/Nobel Prize winners, look at league tables, etc. (I'm English and don't know how the Dutch system works, so apologies if this sounds like nonsense). As an absolute minimum, it MUST be a university which offers Ph.D. degrees in all areas of mathematics.

At good universities, you have nothing to worry about; your lectures/university library will provide everything you need.

Almost certainly, you will buy the wrong books (either far too easy or far too difficult) that you'll never read (or will quickly finish and want to throw away). Almost certainly, your mathematical tastes will change significantly over the next few years.

Sad, but true: many (perhaps even "most" or "almost all") maths books are boring, tedious, unsuitable and impossible to read (and different people will disagree about which books are which, so advice is not much use!)

Personally, almost every mathematical book I bought as an undergraduate or before was a total waste of money; with more experience you'll know better in future what to look for; but now is the worst possible time for you to buy books.

Until you have spent at least one or two years at university, SAVE YOUR MONEY!!! GO TO LIBRARIES!! DO NOT BUY MATHEMATICS BOOKS!!!!! (Unless you are rich or someone else is paying...)

If you do buy books, only do so after you have looked at them for several weeks or months from a library!

Only buy books with LOTS of exercises!(Apart from rare exceptions, e.g. Hardy and Wright: An Introduction to the Theory of Numbers.)

Doing fun research problems of your own, either directly or indirectly inspired by books, is just as important as pure reading; and try not to read too much right now. You won't discover anything new, but that's not the purpose (and the internet is too big - don't look up the answers too quickly, or you'll never develop).

Don't listen to everything physicists say when they're talking about mathematics...! Physicists and mathematicians are very different.

Try looking through things on Wikipedia and the links, and just following wherever your interests take you. You'll learn a surprising amount this way, and enjoy it more too. (But be very careful of the Internet; almost everything online, if not written by proper university mathematicians, is rubbish! MathOverflow and Wikipedia are rare exceptions, but even these have a small amount of rubbish on them).

And finally...

BE FLEXIBLE! Many things you are interested in now might not be to your taste in the future (mathematics is very, very large and no-one can do all parts of it); it is even possible (what a hideous thought!) that you might lose your interest in Mathematics and prefer Physics, Computer Science, Engineering or something else instead; be prepared to change! (Very few people survive up to Ph.D. level and beyond; if you don't become a mathematician, you will get much better jobs after graduation if you do these subjects at university instead...!)

If you still want to study mathematics at university: be prepared for lifetime social exclusion, poverty and unemployment! (This is more for Ph.D. than B.Sc.; perhaps I exaggerate slightly, but you must prepare for the dangers!)

"...lifetime social exclusion,povery and unemployment"?!? Just about everyone I know with a PHD in mathematics has a good paying job and lots of friends,Zen. You won't get rich by any means,but you're making it sound like he's going to end up a hermit mopping floors! Not only is this simply incorrect-unless the mathematician's area is something very abstract and isolated-his life will not be so desolate and deplorable! Now if you refuse to get a job unless it's being on faculty at a university-like some unreasonably stubborn academics-that's a different story.....
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The MathemagicianJul 17 '10 at 2:57

2

I am disgusted by ZH's answer, really. Not only many things are downright wrong ("Don't listen to physicists...", "Prepare for poverty..."), but the very tone of a frustrated rant (with, of course, obligatory exclamations marks and boldface) is anything but inspiring to a beginner.
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Michal KotowskiJul 17 '10 at 10:15

2

That said, Zen, you should really reconsider your last statement. People here (myself included) might not just disagree about your definition of poverty and social exclusion, but find your flippant use of them rather ill-mannered.
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Yemon ChoiJul 18 '10 at 5:14

2

The reason I feel the need to say all this (this is for the benefit of Max and other youngsters) is that what many careers advisors say at school and university, when talking about mathematicians, is just completely wrong. By all means, do Mathematics because you have a lot of interest and ability; but don't do it to get a good job! To get a permanent position, a Ph.D. mathematician needs YEARS of travelling around the world doing various postdocs, only a year or two at a time, with little control over where they go next. It's impossible to have any kind of financial or social stability.
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Zen HarperJul 19 '10 at 17:43

1

Michael: this is getting off the subject of the original question, but what do you mean by "school math", "no extracurricular learning",...I don't understand? I'm wondering what one should do when you've completed a Ph.D. in Pure Mathematics and amassed a modest collection of published papers, but the postdocs/lecturing positions have all dried up and you've left the university system (or are approaching the end of your current position), with no new university job offer: what should you do? In fact, maybe I will ask this question myself...
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Zen HarperJul 19 '10 at 23:16

Hello Max,
let me provide you with yet another opinion:
Imho the best way (and probably also the most fun way) of learning mathematics is not so much by learning mathematical theories from books (you will learn enough theory during your studies anyway) but by actually doing mathematics and by that I mean solving mathematical problems. A good oppertunity to do so is to compete in mathematical competitions (Check out this list from wikipedia http://en.wikipedia.org/wiki/List_of_mathematics_competitions). Not only will it provide you with challenging problems but also it is a good oppertunity in the later stages of the competitions to meet other guys who have the same passion about mathematics as you. It is a great feeling to know that one is not "the only one".
But you were asking for books so I give you my all time favourite:
László Lovász - Combinatorial Problems and Exercises
This books contains nothing but mathematical problems from rather easy ones you can figure out in a quarter of an hour to really challenging ones which could take you up to a week or more to solve. As background you only need basic linear algebra, probability and calculus. Solving all of these problems (of course without looking into the solution before) would probably teach you enough math to start tackling real unsolved problems in the beautiful field of discrete mathematics afterwards, i.e. to do real mathematical research. Sounds tempting doesn't it? ;)

Now that I think about it, I was inspired by two teachers and two books.

The first book was a pretty old one called "Elementary Mathematics" by three Russian authors called G.Dorofeev,Potapov and Rozov. These books were published by a publishing company called Mir Publishers. They used to print extremely good books which were available for a pittance a couple of decades back, but they have shut down now after the collapse of the Soviet Union.

Have you ever been to the Mathematics Institute in Leiden?
You are always welcome. I will show you around
(including the library :)).

As far as I understand, in principle, you can attend some lectures
even in the coming year. I would advise the Algebra course given by Hendrik Lenstra.
The course material is available in Dutch at
http://websites.math.leidenuniv.nl/algebra/

In my opinion, you should not concentrate so much on "doing things properly", but rather read and think about some nice elementary (which is not synonymous to easy) mathematical problem.
There are quite a few such problems. In my last year in high school, I was attending
a "course" on Pell's equation, which was really nice and served the purpose of
doing something outside the standard school curriculum.

First of all, why not go ahead and check out the math library of your university. Go there, sit on the floor and take books of the shelves and just skim through them.

Also, I noticed that you don't mind if there are problems in the books. That's good because you are wasting your time, to be honest, if you don't work through every problem on your own and treat doing the problems as more important than reading the sections.

Alright moving onto specific recomendations:

Conjecture and Proof, M. Lackzovich (this is a great into to doing proofs, has problems, and is wonderful for self study at an amatuer level)

A survey of modern algebra, birkhoff and mclane
(I strongly recommend this book as an intro to modern algebra, it's very clear and easy to work through! Plus it will help you with proofs in a gentle way)

Differential Geometry, stoker
(this is what i learned the subject from, I could not possibly recommend this book more for self study of this subject)

Principles of Analysis, rudin
(this is a 100% no brainier, everyone else recommended it too)

Mechanics, L.D. Landau (course of theoretical physics vol 1)

Introduction to Analysis, whittaker (first edition, the second and third editions are not as good!)

Complex Variables, Fischer

Princeton Lectures on Analysis Vol 1 and 2, Shakarchi and Stein

Introduction to analytic number theory, apostle

Theory of Numbers, serpinski (this guy is from the older school, but its written really really well, and it is always nice to study the masters)

Lectures on Ramanujan, G H Hardy

Disquisitions Arithmatic, Gauss (just try to get as far as you can!)

Abstract Set Theory, Fraenkel (this author has the axioms of set theory named after him, and this book is a really fast and fun read)

Hi! This is a very useful question. Getting the right books is very important. But Math books are expensive, so I suggest you to buy international edition (you can look on abebooks.com) or check your library. Below is a list of books which will definitely prepare you for a rigorous graduate program in Mathematics. It's divided into subfields of Math and within each subfield, it's sorted from easy to hard (from beginning university level to finishing university).

Algebraic Number Theory (Neukirch) -- good book, but you need to at least read the first 4 chapters of Lang's algebra before starting. The section on Analytic Number theory also requires a good understanding of Complex Analysis.

Of course, you don't need to read everything up there. What you need to learn depends on what math you want to do in the future. I think it's good to see a professor and talk to him in person to see what you really want to do.

I found "On Knots" by Louis Kauffman to be very inspiring when I was in high school. It's not a book to be read linearly, but rather you should hop around from section to section. As your mathematical career progresses you'll be able to understand more and more of it.